Rewrite as a simplified fraction. ${2.\overline{21} = {?}}$
Solution: Let $x$ equal the decimal. Set up two equations such that the digits after the decimal point are identical. ${\begin{align*}100x &= 221.2121...\\ 1x &= 2.2121...\end{align*}}$ Subtracting the two equations, we have: ${99x = 219}$ ${x = \dfrac{219}{99}} $ ${x = \dfrac{73}{33}} \text{or} {2 \dfrac{7}{33}}$